Massachusetts Grade 7 Math

Does doubling the radius double everything?

A classroom drawing compares two circular garden beds with radii of 3 meters and 6 meters. The beds are mathematical models with exact circular boundaries. Compare their diameters, circumferences, and areas using pi in the answers. A student predicts that doubling the radius doubles both the boundary length and the area. Decide which parts of the prediction are correct.

Explain the solution

Diameter is twice the radius, so the diameters are 6 and 12 meters. Circumference is 2 times pi times the radius, giving 6 pi and 12 pi meters. The larger circumference is twice the smaller. Area is pi times radius squared, giving 9 pi and 36 pi square meters. The larger area is four times the smaller because doubling the radius multiplies the squared radius by four. The student’s boundary-length prediction is correct, while the area prediction is not. Keep pi in the exact expressions rather than introduce a rounding difference that distracts from the comparison.

Notice the reasoning

Meters measure boundary length; square meters measure covered surface. A formula written without units can hide the distinction. An informal explanation can compare a drawing enlarged by a scale factor of two: lengths grow by two while two-dimensional coverage grows by two in each direction. For a fuller connection, rearrange narrow sectors of a circle toward a parallelogram-like shape whose height approaches the radius and base approaches half the circumference; this motivates area as half the circumference times the radius. A finite cut-out is an approximation, not an exact rectangle. Massachusetts asks for informal relationships as well as using formulas.

Try a related task

Compare a third circle of radius 9 meters with the radius-3 circle. Its radius and circumference are three times as large, while its area is nine times as large. Explain the factors before calculating. Then reverse the question: if a circular model’s area is four times another’s, what radius factor would account for that? Use diagrams and the squared-radius relationship to justify two, with positive radii. This selected comparison supports parts of the state’s circle expectation. It does not replace work defining a circle, measuring circular objects, or developing an informal derivation through classroom models.

Selected references from Massachusetts mathematics standards

The Massachusetts mathematics standards directory links the current mathematics document, adopted in March 2017 and currently listed in the state’s frameworks directory. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

Grade 7 is within the statewide MCAS mathematics assessment range. MCAS is untimed within a regular school day. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration rules or a prediction of a state performance level. Read Massachusetts’s assessment overview.

Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.